#116156

order	512
nilpotency class	3
isoclinism	isoclinic class of #116153
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

(5,7)(9,10,14,20,15,18,13,19)(11,16,24,32,22,33,23,31)(12,25,29,35,27,17,28,36)(21,30,38,39,37,40,26,34);(1,2)(3,7,8,5)(4,6)(9,11)(10,16)(12,26,27,38)(13,23)(14,24)(15,22)(17,34,25,39)(18,33)(19,31)(20,32)(21,28,37,29)(30,35,40,36);(1,3)(2,5)(4,8)(6,7)(9,12,15,27)(10,17,18,25)(11,21,22,37)(13,29,14,28)(16,30,33,40)(19,36,20,35)(23,38,24,26)(31,39,32,34)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,e^b=eh,f^b=fhi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23352);